Riemann surfaces for KPZ with periodic boundaries

TL;DR

This paper explores the complex Riemann surface structure of polylogarithms related to KPZ universality in finite volume, connecting advanced mathematical functions to physical fluctuation phenomena.

Contribution

It introduces a novel geometric framework linking Riemann surfaces of polylogarithms to KPZ fluctuations with periodic boundaries, providing new analytical tools.

Findings

Exact fluctuation results expressed via meromorphic functions on Riemann surfaces

Connections established between KPZ fluctuations and KdV solitons

Insights into large deviations and particle-hole excitations

Abstract

The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of the KPZ height with periodic boundaries are expressed in terms of meromorphic functions on this Riemann surface, summed over all the sheets of a covering map to an infinite cylinder. Connections to stationary large deviations, particle-hole excitations and KdV solitons are discussed.